Proposition 3.10.5.label Let $X$ be a compact topological space, $Y$ be a Hausdorff space, and $f\in C(X;Y)$ be injective, then $f$ is a homeomorphism onto $f(X)$.
Proof. For each $K\suf X$ closed, $K$ and $f(K)$ are compact by Proposition 3.10.3. By Proposition 3.10.4 $f(K)$ is closed. By injectivity $f\inv (f(K))=K$ thus $f\inv$ maps closed sets to closed sets, and hence open sets to open sets.$\square$
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